Plane Geometry - Bruce E. Shapiro
Plane Geometry - Bruce E. Shapiro Plane Geometry - Bruce E. Shapiro
240 SECTION 44. ESTIMATING πFigure 44.1: A square inscribed in a unit circle.Triangle △ABC can be decomposed into two right triangles.Pythagorean theorem we haveFrom the(1 − a) 2 +Solving for a,ÅH22a = 1 − 1 −Also from the Pythagorean theorem,H3 2 = a 2 +Substituting the expression for a,[H3 2 = 1 − 1 −= 1 − 2 1 −+ 1 −ÅH2= 2 − 2 1 −ÅH2ã 2= 1Å ã 2 H22Å ã 2 H222Å ã 2 H222ã 2+Å ã 2 H22ã 2] 2+Å ã 2 H22Å ã 2 H22« CC BY-NC-ND 3.0. Revised: 18 Nov 2012
SECTION 44. ESTIMATING π 241Figure 44.2: One segment of an octagon inscribed in a unit circle is obtainedby spliting each of the triangles in figure 44.1 in half.Therefore (since H 2 = √ 2),By a similar argumentH 2 3 = 2 − 2Ã1 −Ç√ å 22= 2 − √ 22»H 3 = 2 − √ 2 ≈ 0.765367»π 3 = 2 2 H 3 = 4 2 − √ 2 ≈ 3.06147H 2 4 = 2 − 21 −Å ã 2 H32and by induction, we obtainH 2 n = 2 − 21 −Å ã 2 Hn−12so our n th estimate of π is π n = 2 n−1 H n .Our successive estimates of H n also giveRevised: 18 Nov 2012 « CC BY-NC-ND 3.0.
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240 SECTION 44. ESTIMATING πFigure 44.1: A square inscribed in a unit circle.Triangle △ABC can be decomposed into two right triangles.Pythagorean theorem we haveFrom the(1 − a) 2 +Solving for a,ÅH22a = 1 − 1 −Also from the Pythagorean theorem,H3 2 = a 2 +Substituting the expression for a,[H3 2 = 1 − 1 −= 1 − 2 1 −+ 1 −ÅH2= 2 − 2 1 −ÅH2ã 2= 1Å ã 2 H22Å ã 2 H222Å ã 2 H222ã 2+Å ã 2 H22ã 2] 2+Å ã 2 H22Å ã 2 H22« CC BY-NC-ND 3.0. Revised: 18 Nov 2012
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